Gluon propagators in maximally Abelian gauge in SU(3) lattice QCD

In SU(3) lattice QCD, we study diagonal and off-diagonal gluon propagators in the maximally Abelian gauge with $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing. These propagators are studied both in the coordinate space and in the momentum space. The Monte Carlo simulation is performed on ${16}^4$ at $\beta =6.0$ and ${32}^4$ at $\beta =5.8$ and 6.0 at the quenched level. In the four-dimensional Euclidean space-time, the effective mass of diagonal gluons is estimated as $M_{\mathrm{diag}}\simeq 0.3\mathrm{GeV}$ and that of off-diagonal gluons as $M_{\mathrm{off}}\simeq 1\mathrm{GeV}$ in the region of $r=0.4–1.0\mathrm{fm}$. In the momentum space, the effective mass of diagonal gluons is estimated as $M_{\mathrm{diag}}\simeq 0.3\mathrm{GeV}$ and that of off-diagonal gluons as $M_{\mathrm{off}}\simeq 1\mathrm{GeV}$ in the region of $p<1.1\mathrm{GeV}$. The off-diagonal gluon propagator is relatively suppressed in the infrared region and seems to be finite at zero momentum, while the diagonal gluon propagator is enhanced. Furthermore, we also study the functional form of these propagators in momentum space. These propagators are well fitted by $Z/(p^2+m^2{)}^{\nu }$ with fit parameters, $Z$, $m$, and $\nu$ in the region of $p<3.0\mathrm{GeV}$. From the fit results and lattice calculations, all of the spectral functions of diagonal and off-diagonal gluons would have negative regions.

Figure 1

The SU(3) lattice QCD results of gluon propagators $G_{\mu \mu }^{\mathrm{diag}}(r)$ and $G_{\mu \mu }^{\mathrm{off}}(r)$ (top), and their logarithmic plots (bottom) as the function of $r\equiv \sqrt{(x_{\mu }-y_{\mu }{)}^2}$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing in the physical unit. The Monte Carlo simulation is performed on the ${32}^4$ lattice with $\beta =5.8$ and 6.0.

Figure 2

The logarithmic plots of $r^{3/2}{G}_{\mu \mu }^{\mathrm{off}}(r)$ and $r^{3/2}{G}_{\mu \mu }^{\mathrm{diag}}(r)$ as the function of the distance $r$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing, using the SU(3) lattice QCD with ${32}^4$ at $\beta =5.8$ and 6.0.

Figure 3

The logarithmic plots of $r{G}_{\mu \mu }^{\mathrm{off}}(r)$ and $r{G}_{\mu \mu }^{\mathrm{diag}}(r)$ as the function of the four-dimensional Euclidean distance $r$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing with ${32}^4$ at $\beta =5.8$ and 6.0. The solid line denotes the best-fit four-dimensional Yukawa function. For $r{G}_{\mu \mu }^{\mathrm{off}}(r)$, the approximate linear correlation is found for Yukawa-fit mass parameter $m\simeq 1.3\mathrm{GeV}$.

Figure 4

The logarithmic plot of $T^{\mathrm{off}}(p^2)$, $L^{\mathrm{off}}(p^2)$ and $T^{\mathrm{diag}}(p^2)$ as the function of the momentum $p\equiv (p_{\mu }{p}_{\mu }{)}^{1/2}$ with ${16}^4$ (top) and ${32}^4$ (bottom) at $\beta =6.0$. The diagonal gluon propagator is calculated only on ${32}^4$ due to large volume dependence. In the infrared region, the diagonal gluon propagator is largely enhanced, while the longitudinal and transverse parts in the off-diagonal gluon propagator are relatively suppressed. The infrared Abelian dominance is found also in momentum space.

Figure 5

The logarithmic plots of $T^{\mathrm{off}}(p^2)$, $L^{\mathrm{off}}(p^2)$, and $T^{\mathrm{diag}}(p^2)$ as the function of the momentum $p\equiv (p_{\mu }{p}_{\mu }{)}^{1/2}$ with ${32}^4$ at $\beta =6.0$. The transverse and longitudinal parts in the off-diagonal gluon propagator are well described with the four-dimensional Yukawa function, while the diagonal part is not well described in the infrared region. All parts of gluon propagators are well described with the $\nu$ ansatz (30) in the whole region of $p<3.0\mathrm{GeV}$.

Figure 6

The effective mass plot of the longitudinal and transverse parts of off-diagonal gluons, $M_L(t)$ and $M_T(t)$, and the diagonal gluon. The off-diagonal effective masses are obtained on the ${16}^4$ lattice with $\beta =6.0$ using 200 configurations, and the diagonal effective mass is obtained on the ${32}^4$ lattice with $\beta =6.0$ using 40 configurations. Each effective mass $M(t)$ seems to be an increasing function of $t$.

Figure 7

The SU(3) lattice QCD results of $G_{\mu \mu }^{\mathrm{diag}}(r)$ with CD and OD in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing. The Monte Carlo simulation is performed on the ${32}^4$ lattice with $\beta =6.0$. The CD gluon propagator almost coincides with the OD gluon propagator.

Figure 8

The logarithmic plot of $r^{3/2}{G}_{\mu \mu }^{\mathrm{diag}}(r)$ with CD and OD in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing on the ${32}^4$ lattice at $\beta =6.0$. The slopes of these plots in the region of $r=0.4–1.0\mathrm{fm}$ are almost the same.

Figure 9

The logarithmic plots of $T^{\mathrm{diag}}(p^2)$, $L^{\mathrm{diag}}(p^2)$, and $G_{\mu \mu }^{\mathrm{diag}}(p^2)=3T^{\mathrm{diag}}(p^2)+L^{\mathrm{diag}}(p^2)$ as the function of the momentum $p\equiv (p_{\mu }{p}_{\mu }{)}^{1/2}$ in the MA gauge with $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing with ${32}^4$ at $\beta =6.0$. In this gauge, $L^{\mathrm{diag}}(p^2)\ll T^{\mathrm{diag}}(p^2)$ is satisfied at least in the region of $p<3.0\mathrm{GeV}$.